Residually stressed two fibre solids: A spectral approach

Abstract

A nonlinear spectral formulation, which is more general than the traditional classical-invariant formulation, is used to describe the mechanical behaviour of a residually stressed elastic body with two preferred directions (RSTPD); it is hoped that the generality of the spectral formulation facilitates the quest for good constitutive equations for RSTPDs. The strain energy depends on spectral invariants (each with a clear physical meaning) that are functions of the right stretch tensor, residual stress tensor and two preferred-direction-structural tensors; clear meaningful physical invariants are useful in aiding the design of a rigourous experiment to construct a specific form of constitutive equation. An advantage of spectral invariants over classical invariants is illustrated. Finite strain constitutive equations containing single-valued functions, that depend only on a principal stretch, are proposed and their corresponding infinitesimal strain energy functions can be easily obtained from their finite strain counterparts. Using spectral invariants, we easily prove that only 13 of the 37 classical invariants in the corresponding minimal integrity basis are independent; this prove cannot be found in the literature. Some illustrative boundary value results with cylindrical symmetry are given. We also show that the proposed constitutive equations can be easily converted to allow the mechanical influence of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues. The corresponding nearly incompressible strain energy function required for finite element implementation is proposed.